A081285 Table of coefficients of polynomials f_n(q) defined by 1/Product_{i>=1} (1-a q^i)^i = Sum_{n>=0} a^n q^n f_n(q) / ((q)n)^2, where (x)_n is the q-Pochhammer symbol, defined to be Product{i=0..n-1} (1-x q^i).

1, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 2, 4, 2, 4, 2, 4, 2, 1, 0, 1, 1, 0, 1, 2, 4, 6, 7, 8, 12, 12, 14, 12, 12, 8, 7, 6, 4, 2, 1, 0, 1, 1, 0, 1, 2, 4, 6, 12, 12, 21, 26, 37, 40, 55, 52, 61, 60, 61, 52, 55, 40, 37, 26, 21, 12, 12, 6, 4, 2, 1, 0, 1, 1, 0, 1, 2, 4, 6, 12, 18, 26, 38, 57, 76

Offset: 0

Author

Dean Hickerson, using information supplied by Moshe Shmuel Newman and Richard Stanley, Mar 15 2003

Keywords

• nonn
• tabf
• easy

Comments

f_n has degree n(n-1), so n-th row of table has n(n-1)+1 entries. Each row is palindromic. The sum of the terms in the n-th row is n!. The first n+1 terms of the n-th row are the same as the first n terms of A052847.

The 'major index' maj(p) of a permutation p = a_1 a_2 ... a_n is the sum of all i such that a_i > a_(i+1). f_n(q) = Sum_p q^(maj(p)+maj(p^(-1))), where the sum is over all permutations of {1,2,...,n}.

Examples

			f_0 = f_1 = 1, f_2 = 1+q^2, f_3 = 1+q^2+2q^3+q^4+q^6, so sequence begins 1; 1; 1,0,1; 1,0,1,2,1,0,1; ...
		

References

• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; Exercise 4.20.

Links

• Alois P. Heinz, Rows n = 0..40, flattened
• Zhipeng Lu, Symmetric permutation invariants in some tensor products, arXiv:2103.02168 [math.CO], 2021.
• Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Crossrefs

Row sums give A000142.

Cf. A052847.

Programs

Maple

b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
      add(b(u-j, o+j-1)*x^(-u), j=1..u)+
      add(b(u+j-1, o-j)*x^( o), j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n, 0)):
seq(T(n), n=0..10);  # Alois P. Heinz, Apr 28 2018

Mathematica

qpoch[x_, n_] := Product[1-x*q^i, {i, 0, n-1}]; f[0]=1; f[n_] := f[n]=Together[Sum[ -(-1)^r q^Binomial[r, 2] qpoch[q^(n-r+1), r-1]*qpoch[q^(r+1), n-r]/qpoch[q, n-r] f[n-r], {r, 1, n}]]; Join@@Table[CoefficientList[f[n], q], {n, 0, 7}]

Formula

f_n(q) = Sum_{r=1..n} (-1)^(r+1) q^(r(r-1)/2) (q)(n-1) (q)_n / ((q)(r) ((q)(n-r))^2) f(n-r)(q) for n>=1.